packages/misc/limits.rlg

3 limit(sin(x)/x,x,0);
8 limit(sin(x)^2/x,x,0);
13 limit(sin(x)/x,x,1);
18 limit(1/x,x,0);
23 limit(-1/x,x,0);
28 limit((sin(x)-x)/x^3,x,0);
35 limit(x*sin(1/x),x,infinity);
40 limit(sin x/x^2,x,0);
45 limit(x^2*sin(1/x),x,infinity);
53 limit(x^2-6x+4,x,2);
58 limit((x+3)*(2x-1)/(x^2+3x-2),x,-1);
65 limit((sqrt(4+h)-2)/h,h,0);
72 limit((sqrt(x)-2)/(4-x),x,4);
79 limit((x^2-4)/(x-2),x,2);
84 limit(1/(2x-5),x,-1);
91 limit(sqrt(x)/(x+1),x,1);
98 limit((2x+5)/(3x-2),x,infinity);
105 limit((1/(x+3)-2/(3x+5))/(x-1),x,1);
112 limit(sin(3x)/x,x,0);
117 limit((1-cos(x))/x^2,x,0);
124 limit((6x-sin(2x))/(2x+3*sin(4x)),x,0);
131 limit((1-2*cos(x)+cos(2x))/x^2,x,0);
136 limit((3*sin(pi*x) - sin(3*pi*x))/x^3,x,0);
142 limit((cos(a*x)-cos(b*x))/x^2,x,0);
150 limit((e^x-1)/x,x,0);
155 limit((a^x-b^x)/x,x,0);
163 limit(sinh(2x)^2/log(1+x^2),x,0);
168 limit(x^2*(e^(1/x)-1)*(log(x+2)-log(x)),x,infinity);
173 limit(x^alpha*log(x+1)^2/log(x),x,infinity);
178 limit(--------------------,x,infinity)
184 limit((2*cosh(x)-2-x^2)/log(1+x^2)^2,x,0);
191 limit((x*sinh(x)-2+2*cosh(x))/(x^4+2*x^2),x,0);
196 limit((2*sinh(x)-tanh(x))/(e^x-1),x,0);
201 limit(x*tanh(x)/(sqrt(1-x^2)-1),x,0);
206 limit((2*log(1+x)+x^2-2*x)/x^3,x,0);
213 limit((e^(5*x)-2*x)^(1/x),x,0);
219 limit(log(log(x))/log(x)^2,x,infinity);
227 limit((e^x-1)/x, x, 0);
232 limit(((1-x)/log(x))**2, x, 1);
237 limit(x/(e**x-1), x, 0);
244 limit!+(sin(x)/sqrt(x),x,0);
249 limit!-(sin(x)/sqrt(x),x,0);
256 limit(x/log x,x,0);
261 limit(log(1 + x)/log x,x,infinity);
266 limit(log x/sqrt x,x,infinity);
271 limit!+(sqrt x/sin x,x,0);
276 limit(log x,x,0);
281 limit(x*log x,x,0);
286 limit(log x/log(2x),x,0);
291 limit(log x*log(1+x)*(1+x),x,0);
296 limit(log x/x,x,infinity);
301 limit(log x/sqrt x,x,infinity);
306 limit(log x,x,infinity);
311 limit(log(x+1)/sin x,x,0);
316 limit(log(1+1/x)*sin x,x,0);
321 limit(-log(1+x)*(x+2)/sin x,x,0);
326 limit(-log x*(3+x)/log(2x),x,0);
331 limit(log(x+1)^2/sqrt x,x,infinity);
336 limit(log(x + 1) - log x,x,infinity);
341 limit(-(log x)^2/log log x,x,infinity);
346 limit(log(x-1)/sin x,x,0);
352 limit!-(sqrt x/sin x,x,0);
357 limit(log x-log(2x),x,0);
362 limit(sqrt x-sqrt(x+1),x,infinity);
368 limit(sin sin x/x,x,0);
373 limit!-(sin x/cos x,x,pi/2);
378 limit!+(sin x/cos x,x,pi/2);
383 limit(sin x/cosh x,x,infinity);
388 limit(sin x/x,x,infinity);
393 limit(x*sin(1/x),x,0);
398 limit(exp x/((exp x + exp(-x))/2),x,infinity);
403 % limit(exp x/cosh x,x,infinity); % fails in this form, but if cosh is
405 limit((sin(x^2)/(x*sinh x)),x,0);
410 limit(log x*sin(x^2)/(x*sinh x),x,0);
415 limit(sin(x^2)/(x*sinh x*log x),x,0);
420 limit(log x/log(x^2),x,0);
427 limit(log(x^2)-log(x^2+8x),x,0);
432 limit(log(x^2)-log(x^2+8x),x,infinity);
437 limit(sqrt(x+5)-sqrt x,x,infinity);
442 limit(2^(log x),x,0);
449 limit((sin tan x-tan sin x)/(asin atan x-atan asin x),x,0);
456 limit((e+1)^(x^2)/e^x,x,infinity);
462 limit(-----------,x,infinity)
472 limit(log(x-a)/((a-b)*(a-c)) + log(2(x-b))/((b-c)*(b-a))
482 limit(1/(e^x-e^(x-1/x^2)),x,infinity);
488 limit(----------------,x,infinity)
497 limit(x+sqrt x,x,0);
502 limit!+(sqrt x/(x+1),x,0);
507 limit!+(x^(1/3)/(x+1),x,0);
512 limit(log(x)^2/x^(1/3),x,0);
517 limit(log x/x^(1/3),x,0);
532 limit(h,x,0);
543 limit(x^log(1/x),x,infinity);
548 limit(cos x - 1/(e^x^2 - 1),x,0);
553 limit((1+a*x)^(1/x),x,infinity);
557 limit((a*x + 1)   ,x,infinity)
559 limit(x^2*sqrt(4*x^4+5)-2*x^4,x,infinity);
566 limit!+(1/x-1/sin x,x,0);
571 limit(e^(x*sqrt(x^2+1))-e^(x^2),x,infinity);
576 limit( - e   + e              ,x,infinity)
578 limit((e^x+x*log x)/(log(x^4+x+1)+e^sqrt(x^3+1)),x,infinity);
583 limit(---------------------------------,x,infinity)
588 limit!-(1/(x^3-6*x+11*x-6),x,2);
595 limit((x*sqrt(x+5))/(sqrt(4*x^3+1)+x),x,infinity);
602 limit!-(tan x/log cos x,x,pi/2);
616 limit(df(z0,z),z,pi*i/2);
629 limit(df(z1,z),z,pi*i/2);
644 limit(df(z2,z),z,pi/2);
657 limit(df(z3,z),z,pi/2);
673 l0:=limit(f,x,0);
681 l1:=limit(f1,x,0);
689 l2:=limit(f2,x,0);
700 l3:=limit(f3,x,0);
708 l4:=limit(f4,x,0);
716 l5:=limit(f5,x,0);
727 l6:=limit(f6,x,0);
735 l7:=limit(f7,x,0);
743 l8:=limit(f8,x,0);
752 limit(log(1+x)^2/x^(1/3),x,infinity);
757 limit(e^(log(1+x)^2/x^(1/3)),x,infinity);
772 limit(ss,x,0);
777 limit(exp(ss),x,0);
784 limit(log x,x,-1);
789 limit(log(ss),x,0);
804 limit(ss,x,0);
819 limit(h,x,0);
829 COMMENT one sided limit from bug report by Zbynek Konecny ;
832 limit!-((abs(x)-3)/(x-3),x,3);
841 limit((11/5)^x,x,infinity);
846 limit((2/5)^x,x,infinity);
851 limit(2.1^x,x,infinity);
856 limit(2.1^(1/x),x,0);
865 limit((11/5)^x,x,infinity);
870 limit((2/5)^x,x,infinity);
875 limit(2.1^x,x,infinity);
880 limit(2.1^(1/x),x,0);
889 limit(log(2)*log(x),x,infinity);
898 limit(log(2)*log(x),x,infinity);